On the radicality property for spaces of symbols of bounded Volterra operators

In a recent paper of the authors together with A. Aleman, it is shown that the Bloch space B in the unit disc has the following radicality property: if an analytic function g satisfies that gⁿ∈ B, then gᵐ∈ B, for all m≤ n. Since B coincides with the space T(Aᵖ_α) of analytic symbols g such that the Volterra-type operator Tgf(z)= ∫₀ᶻ f(ζ)g'(ζ)\,dζ is bounded on the classical weighted Bergman space Aᵖ_α, the radicality property was used to study the composition of paraproducts Tg and Sgf=Tfg on Aᵖα. Motivated by this fact, we prove that T(Aᵖ_ω) also has the radicality property, for any radial weight ω. Unlike the classical case, the lack of a precise description of T(Aᵖ_ω) for a general radial weight, induces us to prove the radicality property for Aᵖ_ω from precise norm-operator results for compositions of analytic paraproducts.